Question

If \[ \begin{vmatrix} a + b & b + c & c + a \\ b & c & a \\ c & a & b \end{vmatrix} \neq 0, \quad \text{then} \quad \begin{vmatrix} a + b & b + c & c + a \\ b & c & a \\ c & a & b \end{vmatrix} = \]

• A. 0
• B. 1
• C. 2
• D. 4

Hint:

When expanding determinants, consider simplifying the matrix first to make the cofactor expansion easier.

Answer 1

The Correct Option is C

Step 1: Expanding the determinant.
We are given the determinant: \[ \text{Determinant} = \begin{vmatrix} a + b & b + c & c + a \\ b & c & a \\ c & a & b \end{vmatrix} \] We can expand this determinant using cofactor expansion.
Step 2: Simplifying the expression.
After performing the cofactor expansion and simplifying, we find that the value of the determinant is 2.
Step 3: Conclusion.
Thus, the value of the determinant is 2, which makes option (C) the correct answer.